How would an ordinary ~$20 knife from a sporting good store compare to a knife from say the 1300 A.D. in terms of quality, workmanship, and how well it could hold an edge?

First, a word of warning-my own specialty of weapons of war, and I have not been able to find much in the secondary literature about late medieval knives specifically - as opposed to armour and swords, which have been dealt with extensively. This is thus a more general answer than I would prefer, about medieval ferrous metallurgy in general, and the way it would apply to knife manufacture. You may find my previous answer about ‘good steel’ in medieval armour and weapons helpful.

As others have noted, the salient difference between a modern knife from a sporting goods store and a 14th century knife is reliability. You basically know what you are getting when it comes to the knife from the sporting goods store - some alloy of steel and chromium, probably (since so many knives in this price range are stainless steel) - indeed many knives advertise their exact alloy composition. You can also expect that it will be essentially homogenous throughout - that the composition of the alloys will be the same throughout. This is because modern steelmaking processes allow nearly exact control of alloy composition, and can ensure uniformity.

The 14th century knife, on the other hand, presents a different picture. Nearly every phase of the manufacture of a medieval knife would be subject to greater uncertainty than modern cutlery. To quote myself from a previous answer:

To lay out the basic process, to smelt iron from iron ore (which you need to do to get iron/steel, unless you are making an item from a meteorite) you need to heat the ore until the various non-iron bits combine with oxygen (burn off, basically) or melt out. The simplest and earliest way to do this is in a bloomery, where the iron itself never melts but forms a kind of spongey mass of variable composition, a bloom. Alternately, if you make the furnace hot enough you can melt the iron, which produces a high-carbon mass bit of molten ferrous material called cast iron or pig iron. Blast furnaces were invented in 13th century Europe, though the main source of 14th century steel and iron would be bloomeries.

Good quality blades would be made of steel. This can be made in a bloomery under the right conditions. Now, the entire weapon or tool does not need to be made of steel. For some items (swords/knifes/spearheads, files) only the edge or perhaps the surface will need to be made of steel. In order to do this there are two means. The first is to weld a small bit of steel to iron, like making the body of an axe out of iron and then forge-welding a steel blade to it. This is used in traditional Japanese swordmaking, and there is ample evidence for it in medieval European tools and weapons. Some Celtic and Roman swords were made in this way, though the smiths didn't consistently put their hardes material on the edges. The other way of making something partially steel is to case-carburize or 'case harden' it - covering the iron item in a carbon source like plant or animal matter or charcoal and heating it in a low-oxygen environment. Eventually, the iron absorbs carbon and the surface becomes steel.

|In order to make a steel object harder you need to heat treat it by plunging heated medium carbon (.3% or better) steel into a liquid. This changes the structure of the material as it cools rapidly - it forms different crystaline structures - most notably Martensite, a very hard crystalline structure of iron and iron carbide. Now you can plunge it into cold water, (a 'full quench') which produces fantastically hard steel (up to 800 on the vickers hardness scale), to the point where it is brittle. This steel then needs to be re-heated or tempered to reduce its hardness by a little bit and increase its toughness by a lot (so that it is around 380-450 on the vickers hardness scale). Or you can either wait to quench the steel, or quench it into a hot liquid, or interrupt the quench - this is a 'slack quench' - and it hardens the steel somewhat, but not as much. Steels hardened this way are only a little harder than high-tin bronze (like 320 on the Vickers hardness scale compared with 250). Slack quenching was used earlier and more often than full quenching - it is considered an 'inferior' process today, but the lack of extra steps and the perception that it was 'safer' for the object made it more popular (fewer steps also meant there were fewer places where something could go wrong).

There is also the special case of phosphoric iron is iron made from ore with phosphorous impurities in it (bog iron has phosphorous in some cases). This makes it harder - around as hard as a low-tin bronze (up to 200 vickers). Small items can be made from phosphoric iron - a number of medieval 'bodkin' arrowheads are made from phosphoric iron. Unlike carbon steel, phosphoric iron doesn't get harder with heat treatment, and so tops out around 200 vickers hardness or so.

Now, every step of this process - making iron/steel in a bloomery, making that bloom into an object, and then heat treating the (nearly finished) object introduced uncertainty. The original bloom may have uneven carbon content, or include unpredictable slag inclusions. This would make the knife unevenly hard, giving it possible points of failure. While making the bloom into a knife, the smith might introduce additional weaknesses during forging (on weld lines, for instance, if the knife was made in multiple pieces). If the knife was case-hardened, it may be difficult to determine the final carbon content, leaving a knife that was either brittle or still too soft. Finally, heat treatment could either produce a knife that was overtempered - basically as soft as it was when heat treatment began - or too hard if tempering was not successful, leaving the knife prone to breaking. In every step, smiths had only their own eyes and imprecise measurments to gauge the complex chemical processes occurring. It could be hard to tell when things went right, and when they went wrong.

This is not to say that knives were a total crapshoot. People knew that some steels were better than others - there was a trade in steel (in the form of plates or ingots) from Styria in Austria (largely for weapons and armour) and iron from Sweden (in the form of small ‘osmunds’ of iron), and a corresponding trade in finished goods made with these ‘high end’ raw materials. Iron from the iron-producing Ariege region of the French Pyrennes finds its way into goods from Narbonne, while Vicdossis within the Ariege region imported high quality knives made in Parma, in northern Italy. If anything, paying for good quality in medieval steel could produce more pronounced results than paying for good quality in modern metal goods - while modern cheap knives may be bad at holding an edge, they aren’t as likely to fail spectacularly. People would pay for quality, and poor goods could be bought at a discount, to quote the 15th century German preacher Johannes Nider said: "Every person intends to do his craft (hantwerk) faithfully. If a thing is not good, it sells much more cheaply." With that said, telling quality was harder then than it is now, the the overall variability was much higher.

So to conclude, a knife in the 14th century might hold an edge quite well while being quite tough (if it was made with heat treated medium-carbon steel). Or it might be brittle, or soft.The high end would be around what a modern knife is, while the low end would be a tool that we would consider rather flawed, but would most of the time be good enough to do the job (unless is broke).

Sources: Williams, Alan - The Sword and the Crucible

And

Williams, Alan - The Knight and the Blast Furnace

S. Leroy and Others - The medieval iron market in Ariège (France). Multidisciplinary analytical approach and multivariate analyses - Journal of Archaelogical Science Vol. 39, 2012